Progetto
Andrea Corvaglia
17 agosto 2019
TO DO:
La Gpu con 112 livelli è inutilizzabile, possiamo dividere in integrata e dedicata, oppure dividere per le marche.
@Dario: Per la gpu farei innanzitutto una divisione tra integrata e dedicata, e tra quelle dedicate le suddividerei per VRAM (inizialmente pensavo di cercare dei prezzi per avere una stima quantitativa di quanto valga la gpu dedicata, però bisognerebbe cercare prezzi medi e visto che i modelli son tanti mi pare troppo lavoro e non chissà quanto significativo)
Anche la memoria è problematica, ho già estratto la presenza o meno dell’SSD, ma bisogna pensare ad un modo di valutare la memoria.
@Dario: ottimo la divisione SSD/HD, per avere un indicazione quantitativa potremmo cercare il prezzo di un gb di SSD e di un gb di HD da una fonte affidabile, e pesare l’importanza delle dimensioni su questo indicatore, che ne dite?
data <- read.csv("../data/Laptop2.csv")
str(data)## 'data.frame': 1303 obs. of 17 variables:
## $ X : int 1 2 3 4 5 6 7 8 9 10 ...
## $ Company : Factor w/ 19 levels "Acer","Apple",..: 2 2 8 2 2 1 2 2 3 1 ...
## $ Product : Factor w/ 618 levels "110-15ACL (A6-7310/4GB/500GB/W10)",..: 302 300 51 302 302 59 302 300 615 431 ...
## $ TypeName : Factor w/ 6 levels "2 in 1 Convertible",..: 5 5 4 5 5 4 5 5 5 5 ...
## $ Inches : num 13.3 13.3 15.6 15.4 13.3 15.6 15.4 13.3 14 14 ...
## $ ScreenResolution: Factor w/ 40 levels "1366x768","1440x900",..: 24 2 9 26 24 1 26 2 9 16 ...
## $ Cpu : Factor w/ 118 levels "AMD A10-Series 9600P 2.4GHz",..: 55 53 64 75 57 15 74 53 96 73 ...
## $ Ram : int 8 8 8 16 8 4 16 8 16 8 ...
## $ Memory : Factor w/ 39 levels "1.0TB HDD","1.0TB Hybrid",..: 5 3 17 30 17 27 16 16 30 17 ...
## $ Gpu : Factor w/ 110 levels "AMD FirePro W4190M",..: 59 52 54 10 60 18 61 52 98 62 ...
## $ OpSys : Factor w/ 9 levels "Android","Chrome OS",..: 5 5 6 5 5 7 4 5 7 7 ...
## $ Weight : num 1.37 1.34 1.86 1.83 1.37 2.1 2.04 1.34 1.3 1.6 ...
## $ Price : num 1340 899 575 2537 1804 ...
## $ Frequenza : num 2.3 1.8 2.5 2.7 3.1 3 2.2 1.8 1.8 1.6 ...
## $ Risoluzione : Factor w/ 15 levels "1366x768","1440x900",..: 11 2 4 13 11 1 13 2 4 4 ...
## $ Pixel : int 4096000 1296000 2073600 5184000 4096000 1049088 5184000 1296000 2073600 2073600 ...
## $ SolidStateDisk : Factor w/ 2 levels "False","True": 2 1 2 2 2 1 1 1 2 2 ...head(data)## X Company Product TypeName Inches
## 1 1 Apple MacBook Pro Ultrabook 13.3
## 2 2 Apple Macbook Air Ultrabook 13.3
## 3 3 HP 250 G6 Notebook 15.6
## 4 4 Apple MacBook Pro Ultrabook 15.4
## 5 5 Apple MacBook Pro Ultrabook 13.3
## 6 6 Acer Aspire 3 Notebook 15.6
## ScreenResolution Cpu Ram
## 1 IPS Panel Retina Display 2560x1600 Intel Core i5 2.3GHz 8
## 2 1440x900 Intel Core i5 1.8GHz 8
## 3 Full HD 1920x1080 Intel Core i5 7200U 2.5GHz 8
## 4 IPS Panel Retina Display 2880x1800 Intel Core i7 2.7GHz 16
## 5 IPS Panel Retina Display 2560x1600 Intel Core i5 3.1GHz 8
## 6 1366x768 AMD A9-Series 9420 3GHz 4
## Memory Gpu OpSys Weight
## 1 128GB SSD Intel Iris Plus Graphics 640 macOS 1.37
## 2 128GB Flash Storage Intel HD Graphics 6000 macOS 1.34
## 3 256GB SSD Intel HD Graphics 620 No OS 1.86
## 4 512GB SSD AMD Radeon Pro 455 macOS 1.83
## 5 256GB SSD Intel Iris Plus Graphics 650 macOS 1.37
## 6 500GB HDD AMD Radeon R5 Windows 10 2.10
## Price Frequenza Risoluzione Pixel SolidStateDisk
## 1 1339.69 2.3 2560x1600 4096000 True
## 2 898.94 1.8 1440x900 1296000 False
## 3 575.00 2.5 1920x1080 2073600 True
## 4 2537.45 2.7 2880x1800 5184000 True
## 5 1803.60 3.1 2560x1600 4096000 True
## 6 400.00 3.0 1366x768 1049088 Falsesummary(data)## X Company Product
## Min. : 1.0 Dell :297 XPS 13 : 30
## 1st Qu.: 331.5 Lenovo :297 Inspiron 3567 : 29
## Median : 659.0 HP :274 250 G6 : 21
## Mean : 660.2 Asus :158 Legion Y520-15IKBN: 19
## 3rd Qu.: 990.5 Acer :103 Vostro 3568 : 19
## Max. :1320.0 MSI : 54 Inspiron 5570 : 18
## (Other):120 (Other) :1167
## TypeName Inches
## 2 in 1 Convertible:121 Min. :10.10
## Gaming :205 1st Qu.:14.00
## Netbook : 25 Median :15.60
## Notebook :727 Mean :15.02
## Ultrabook :196 3rd Qu.:15.60
## Workstation : 29 Max. :18.40
##
## ScreenResolution
## Full HD 1920x1080 :507
## 1366x768 :281
## IPS Panel Full HD 1920x1080 :230
## IPS Panel Full HD / Touchscreen 1920x1080: 53
## Full HD / Touchscreen 1920x1080 : 47
## 1600x900 : 23
## (Other) :162
## Cpu Ram
## Intel Core i5 7200U 2.5GHz :190 Min. : 2.000
## Intel Core i7 7700HQ 2.8GHz:146 1st Qu.: 4.000
## Intel Core i7 7500U 2.7GHz :134 Median : 8.000
## Intel Core i7 8550U 1.8GHz : 73 Mean : 8.382
## Intel Core i5 8250U 1.6GHz : 72 3rd Qu.: 8.000
## Intel Core i5 6200U 2.3GHz : 68 Max. :64.000
## (Other) :620
## Memory Gpu
## 256GB SSD :412 Intel HD Graphics 620 :281
## 1TB HDD :223 Intel HD Graphics 520 :185
## 500GB HDD :132 Intel UHD Graphics 620 : 68
## 512GB SSD :118 Nvidia GeForce GTX 1050: 66
## 128GB SSD + 1TB HDD: 94 Nvidia GeForce GTX 1060: 48
## 128GB SSD : 76 Nvidia GeForce 940MX : 43
## (Other) :248 (Other) :612
## OpSys Weight Price Frequenza
## Windows 10:1072 Min. :0.690 Min. : 174 Min. :0.900
## No OS : 66 1st Qu.:1.500 1st Qu.: 599 1st Qu.:2.000
## Linux : 62 Median :2.040 Median : 977 Median :2.500
## Windows 7 : 45 Mean :2.039 Mean :1124 Mean :2.299
## Chrome OS : 27 3rd Qu.:2.300 3rd Qu.:1488 3rd Qu.:2.700
## macOS : 13 Max. :4.700 Max. :6099 Max. :3.600
## (Other) : 18
## Risoluzione Pixel SolidStateDisk
## 1920x1080:841 Min. :1049088 False:460
## 1366x768 :308 1st Qu.:1440000 True :843
## 3840x2160: 43 Median :2073600
## 3200x1800: 27 Mean :2168807
## 1600x900 : 23 3rd Qu.:2073600
## 2560x1440: 23 Max. :8294400
## (Other) : 38nums <- sapply(data, is.numeric)
var_numeric <- data[,nums]
head(var_numeric)## X Inches Ram Weight Price Frequenza Pixel
## 1 1 13.3 8 1.37 1339.69 2.3 4096000
## 2 2 13.3 8 1.34 898.94 1.8 1296000
## 3 3 15.6 8 1.86 575.00 2.5 2073600
## 4 4 15.4 16 1.83 2537.45 2.7 5184000
## 5 5 13.3 8 1.37 1803.60 3.1 4096000
## 6 6 15.6 4 2.10 400.00 3.0 1049088data$Weight<-as.numeric(data$Weight)
data$Ram<-as.numeric(data$Ram)sapply(data, function(x)(sum(is.na(x))))## X Company Product TypeName
## 0 0 0 0
## Inches ScreenResolution Cpu Ram
## 0 0 0 0
## Memory Gpu OpSys Weight
## 0 0 0 0
## Price Frequenza Risoluzione Pixel
## 0 0 0 0
## SolidStateDisk
## 0# Non ci sono missing data!plot(data$Company,data$Price)
class(data$Ram)## [1] "numeric"plot(density(data$Frequenza))
#hist(data$Price, breaks=25, probability=TRUE)
#lines(density(data$Price))library(ggplot2)## Registered S3 methods overwritten by 'ggplot2':
## method from
## [.quosures rlang
## c.quosures rlang
## print.quosures rlangggplot(data,aes(x = Price)) +
geom_histogram(aes(y =..density..),
bins= 25,
fill = "grey",
color ="black") +
geom_vline(xintercept = quantile(data$Price, 0.50), color = "dark red", lty = 2) +
geom_vline(xintercept = mean(data$Price), color = "dark blue", lty = 2) +
labs(x = "Price", y ="Density") +
ggtitle("Price Distribution with mean and median") +
geom_density()
Quite skewed to the right, mean > media
We could try to apply a correction like Log(Y)
data$LogPrice=log(data$Price)
ggplot(data,aes(x = log(Price))) +
geom_histogram(aes(y =..density..),
bins= 25,
fill = "grey",
color ="black") +
geom_vline(xintercept = quantile(data$LogPrice, 0.50), color = "dark red", lty = 2) +
geom_vline(xintercept = mean(data$LogPrice), color = "dark blue", lty = 2) +
labs(x = "log(Price)", y ="Density") +
ggtitle("log(Price) Distribution with mean and median")+ geom_density()
Now the distribution is looking a bit better (as regards normality)
ggplot(data,aes(x = Price)) +
geom_histogram(aes(y =..density..),
bins= 25,
fill = "grey",
color ="black") +
geom_vline(xintercept = mean(data$Price), color = "dark red") +
geom_vline(xintercept = mean(data$Price) + sd(data$Price), color = "dark red", lty = 2) +
geom_vline(xintercept = mean(data$Price) - sd(data$Price), color = "dark red", lty = 2) +
labs(x = "Price", y ="Density") +
ggtitle("Price Distribution (mean +/- sd)") +
geom_density()
ggplot(data,aes(x = log(Price))) +
geom_histogram(aes(y =..density..),
bins= 25,
fill = "grey",
color ="black") +
geom_vline(xintercept = mean(data$LogPrice), color = "dark red") +
geom_vline(xintercept = mean(data$LogPrice) + sd(data$LogPrice), color = "dark red", lty = 2) +
geom_vline(xintercept = mean(data$LogPrice) - sd(data$LogPrice), color = "dark red", lty = 2) +
labs(x = "log(Price)", y ="Density") +
ggtitle("log(Price) Distribution (mean +/- sd)") +
geom_density()
ggplot(data,aes(x = Price)) +
geom_histogram(aes(y =..density..),
bins= 25,
fill = "grey",
color ="black") +
geom_vline(xintercept = quantile(data$Price, 0.25), color = "dark red",lty = 2) +
geom_vline(xintercept = quantile(data$Price, 0.5), color = "dark red", ) +
geom_vline(xintercept = quantile(data$Price, 0.75), color = "dark red", lty = 2) +
labs(x = "Price", y ="Density") +
ggtitle("Price Distribution (quartiles)") +
geom_density()
ggplot(data,aes(x = log(Price))) +
geom_histogram(aes(y =..density..),
bins= 25,
fill = "grey",
color ="black") +
geom_vline(xintercept = quantile(data$LogPrice, 0.25), color = "dark red",lty = 2) +
geom_vline(xintercept = quantile(data$LogPrice, 0.5), color = "dark red", ) +
geom_vline(xintercept = quantile(data$LogPrice, 0.75), color = "dark red", lty = 2) +
labs(x = "log(Price)", y ="Density") +
ggtitle("log(Price) Distribution (quartiles)") +
geom_density()
Descrittive variabile dipendente price
ggplot(data, aes(x = Price, fill = TypeName)) +
geom_density(size = 0.6, alpha = .3) +
labs(x = "Price", y ="Density", fill = "TypeName") +
ggtitle("Price Density Distribution For TypeName") 
ggplot(data, aes(x = log(Price), fill = TypeName)) +
geom_density(size = 0.6, alpha = .3) +
labs(x = "log(Price)", y ="Density", fill = "TypeName") +
ggtitle("log(Price) Density Distribution For TypeName") 
ggplot(data, aes(x = Price, fill = SolidStateDisk)) +
geom_density(size = 0.6, alpha = .3) +
labs(x = "Price", y ="Density", fill = "SolidStateDisk") +
ggtitle("Price Density Distribution For SolidStateDisk") 
ggplot(data, aes(x = log(Price), fill = SolidStateDisk)) +
geom_density(size = 0.6, alpha = .3) +
labs(x = "log(Price)", y ="Density", fill = "SolidStateDisk") +
ggtitle("log(Price) Density Distribution For SolidStateDisk") 
library(psych)##
## Attaching package: 'psych'## The following objects are masked from 'package:ggplot2':
##
## %+%, alphadescribe(data$Price)## vars n mean sd median trimmed mad min max range skew
## X1 1 1303 1123.69 699.01 977 1038.47 619.73 174 6099 5925 1.52
## kurtosis se
## X1 4.34 19.36library(nortest)
# NORMALITA'
boxplot(data$Price)
qqnorm(data$Price);qqline(data$Price)
shapiro.test(data$Price)##
## Shapiro-Wilk normality test
##
## data: data$Price
## W = 0.89382, p-value < 2.2e-16ad.test(data$Price)##
## Anderson-Darling normality test
##
## data: data$Price
## A = 28.319, p-value < 2.2e-16#wilcox.test(data$Price, conf.int = TRUE, mu = ) #worth it?
#if(!require(Envstats)) install.packages("EnvStats")
library(EnvStats)##
## Attaching package: 'EnvStats'## The following objects are masked from 'package:stats':
##
## predict, predict.lm## The following object is masked from 'package:base':
##
## print.defaultvarTest(sample(data$Price), sigma.squared = (sd(data$Price)*sd(data$Price)))##
## Chi-Squared Test on Variance
##
## data: sample(data$Price)
## Chi-Squared = 1302, df = 1302, p-value = 0.9896
## alternative hypothesis: true variance is not equal to 488613.6
## 95 percent confidence interval:
## 453149.5 528432.0
## sample estimates:
## variance
## 488613.6Trying with the log correction:
# Correzione NORMALITA'
library(nortest)
boxplot(data$LogPrice)
qqnorm(data$LogPrice);qqline(data$LogPrice)
shapiro.test(data$LogPrice) #better than before, but still not normal according to shapiro##
## Shapiro-Wilk normality test
##
## data: data$LogPrice
## W = 0.99252, p-value = 3.628e-06ad.test(data$LogPrice)##
## Anderson-Darling normality test
##
## data: data$LogPrice
## A = 2.5942, p-value = 1.515e-06T-test
# One sample
ref <- mean(data$Price)
Apple<-data$Price[data$Company=="Apple"]
t.test(Apple,mu=ref,alternative = "greater")##
## One Sample t-test
##
## data: Apple
## t = 3.5944, df = 20, p-value = 0.000906
## alternative hypothesis: true mean is greater than 1123.687
## 95 percent confidence interval:
## 1352.823 Inf
## sample estimates:
## mean of x
## 1564.199# Wilcoxon Signed Rank Test
wilcox.test(Apple, mu=ref, conf.int = TRUE)##
## Wilcoxon signed rank test
##
## data: Apple
## V = 206, p-value = 0.0008516
## alternative hypothesis: true location is not equal to 1123.687
## 95 percent confidence interval:
## 1234.50 1829.26
## sample estimates:
## (pseudo)median
## 1514.275#Two sample
Other <-data$Price[data$Company!="Apple"]
wilcox.test(Apple, Other, alternative = "g")##
## Wilcoxon rank sum test with continuity correction
##
## data: Apple and Other
## W = 19689, p-value = 0.0001358
## alternative hypothesis: true location shift is greater than 0# F test sulla varianza
var.test(Apple, Other, alternative = "two.sided")##
## F test to compare two variances
##
## data: Apple and Other
## F = 0.64574, num df = 20, denom df = 1281, p-value = 0.2401
## alternative hypothesis: true ratio of variances is not equal to 1
## 95 percent confidence interval:
## 0.3755878 1.3509884
## sample estimates:
## ratio of variances
## 0.6457382Variabili qualitative: tabella di contingenza e chi quadro
b<-data
b.table<-table(b$SolidStateDisk,b$TypeName)
b.table##
## 2 in 1 Convertible Gaming Netbook Notebook Ultrabook Workstation
## False 26 27 13 372 16 6
## True 95 178 12 355 180 23prop.table(b.table,2)##
## 2 in 1 Convertible Gaming Netbook Notebook Ultrabook
## False 0.21487603 0.13170732 0.52000000 0.51169188 0.08163265
## True 0.78512397 0.86829268 0.48000000 0.48830812 0.91836735
##
## Workstation
## False 0.20689655
## True 0.79310345# chi square test
chisq.test(b.table)##
## Pearson's Chi-squared test
##
## data: b.table
## X-squared = 203.18, df = 5, p-value < 2.2e-16chi=chisq.test(b.table)
chi_norm=chi$statistic/(nrow(b)*min(nrow(b.table)-1,ncol(b.table)-1))
chi_norm## X-squared
## 0.1559288summary(b.table)## Number of cases in table: 1303
## Number of factors: 2
## Test for independence of all factors:
## Chisq = 203.18, df = 5, p-value = 5.944e-42Correlazione per variabili quantitative
# seleziona solo variabili quantitative
nums <- sapply(data, is.numeric)
var_numeric <- data[,nums]
head(var_numeric)## X Inches Ram Weight Price Frequenza Pixel LogPrice
## 1 1 13.3 8 1.37 1339.69 2.3 4096000 7.200194
## 2 2 13.3 8 1.34 898.94 1.8 1296000 6.801216
## 3 3 15.6 8 1.86 575.00 2.5 2073600 6.354370
## 4 4 15.4 16 1.83 2537.45 2.7 5184000 7.838915
## 5 5 13.3 8 1.37 1803.60 3.1 4096000 7.497540
## 6 6 15.6 4 2.10 400.00 3.0 1049088 5.991465var_numeric$X=NULL# Matrice di correlazione
R<-cor(var_numeric)
R## Inches Ram Weight Price Frequenza
## Inches 1.00000000 0.2379928 0.82763110 0.06819667 0.3078698
## Ram 0.23799280 1.0000000 0.38387409 0.74300714 0.3680005
## Weight 0.82763110 0.3838741 1.00000000 0.21036980 0.3204336
## Price 0.06819667 0.7430071 0.21036980 1.00000000 0.4302931
## Frequenza 0.30786980 0.3680005 0.32043359 0.43029310 1.0000000
## Pixel -0.08639917 0.3963585 -0.04403379 0.51548639 0.1352935
## LogPrice 0.04432871 0.6848033 0.15167383 0.92758068 0.5041461
## Pixel LogPrice
## Inches -0.08639917 0.04432871
## Ram 0.39635848 0.68480333
## Weight -0.04403379 0.15167383
## Price 0.51548639 0.92758068
## Frequenza 0.13529350 0.50414608
## Pixel 1.00000000 0.48490475
## LogPrice 0.48490475 1.00000000# Test di correlazione. (Spearsman's o Kendall tau)
cor.test(var_numeric$Inches, var_numeric$Weight)##
## Pearson's product-moment correlation
##
## data: var_numeric$Inches and var_numeric$Weight
## t = 53.187, df = 1301, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.8097181 0.8440031
## sample estimates:
## cor
## 0.8276311#corrgram(var_numeric)
# Correlazione come grafo
library(qgraph)## Registered S3 methods overwritten by 'huge':
## method from
## plot.sim BDgraph
## print.sim BDgraphdetcor=cor(as.matrix(var_numeric), method="pearson")
round(detcor, 2)## Inches Ram Weight Price Frequenza Pixel LogPrice
## Inches 1.00 0.24 0.83 0.07 0.31 -0.09 0.04
## Ram 0.24 1.00 0.38 0.74 0.37 0.40 0.68
## Weight 0.83 0.38 1.00 0.21 0.32 -0.04 0.15
## Price 0.07 0.74 0.21 1.00 0.43 0.52 0.93
## Frequenza 0.31 0.37 0.32 0.43 1.00 0.14 0.50
## Pixel -0.09 0.40 -0.04 0.52 0.14 1.00 0.48
## LogPrice 0.04 0.68 0.15 0.93 0.50 0.48 1.00# plot corr matrix: green positive red negative
qgraph(detcor, shape="circle", posCol="darkgreen", negCol="darkred")
Boxplot di confronto (pre-anova)
boxplot(data$Price~data$Company,
main="Boxplot Prezzo per compagnia",
col= rainbow(6),
horizontal = F)
boxplot(data$Price~data$SolidStateDisk,
main="Prezzo vs ssd",
col= rainbow(2),
horizontal = F)
ANOVA
A una via
lmA = lm(Price ~ SolidStateDisk, data=data)
summary(lmA)##
## Call:
## lm(formula = Price ~ SolidStateDisk, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1214.8 -343.3 -96.8 261.1 4710.2
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 637.86 27.98 22.80 <2e-16 ***
## SolidStateDiskTrue 750.93 34.78 21.59 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 600 on 1301 degrees of freedom
## Multiple R-squared: 0.2638, Adjusted R-squared: 0.2632
## F-statistic: 466.2 on 1 and 1301 DF, p-value: < 2.2e-16drop1(lmA, test = 'F')## Single term deletions
##
## Model:
## Price ~ SolidStateDisk
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 468355897 16672
## SolidStateDisk 1 167819064 636174961 17069 466.17 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(lmA)## Analysis of Variance Table
##
## Response: Price
## Df Sum Sq Mean Sq F value Pr(>F)
## SolidStateDisk 1 167819064 167819064 466.17 < 2.2e-16 ***
## Residuals 1301 468355897 359997
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1library(lsmeans)## Loading required package: emmeans## The 'lsmeans' package is now basically a front end for 'emmeans'.
## Users are encouraged to switch the rest of the way.
## See help('transition') for more information, including how to
## convert old 'lsmeans' objects and scripts to work with 'emmeans'.ls_SolidStateDisk = lsmeans(lmA,pairwise ~ SolidStateDisk,adjust = 'tukey')
ls_SolidStateDisk$contrasts ## contrast estimate SE df t.ratio p.value
## False - True -751 34.8 1301 -21.591 <.0001ls_SolidStateDisk$lsmeans## SolidStateDisk lsmean SE df lower.CL upper.CL
## False 638 28.0 1301 583 693
## True 1389 20.7 1301 1348 1429
##
## Confidence level used: 0.95plot(ls_SolidStateDisk$lsmeans, alpha = .05)
lmB = lm(Price ~ Company, data=data)
summary(lmB)##
## Call:
## lm(formula = Price ~ Company, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2317.1 -452.8 -127.4 288.5 3812.6
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 626.78 63.43 9.881 < 2e-16 ***
## CompanyApple 937.42 154.14 6.082 1.57e-09 ***
## CompanyAsus 477.39 81.53 5.856 6.03e-09 ***
## CompanyChuwi -312.48 377.06 -0.829 0.407416
## CompanyDell 559.29 73.62 7.597 5.80e-14 ***
## CompanyFujitsu 102.22 377.06 0.271 0.786352
## CompanyGoogle 1050.89 377.06 2.787 0.005397 **
## CompanyHP 441.00 74.41 5.927 3.96e-09 ***
## CompanyHuawei 797.22 459.62 1.735 0.083065 .
## CompanyLenovo 459.61 73.62 6.243 5.81e-10 ***
## CompanyLG 1472.22 377.06 3.904 9.93e-05 ***
## CompanyMediacom -331.78 251.46 -1.319 0.187270
## CompanyMicrosoft 985.53 270.37 3.645 0.000278 ***
## CompanyMSI 1102.13 108.16 10.190 < 2e-16 ***
## CompanyRazer 2719.37 251.46 10.814 < 2e-16 ***
## CompanySamsung 786.67 223.77 3.515 0.000454 ***
## CompanyToshiba 641.04 112.51 5.698 1.50e-08 ***
## CompanyVero -409.35 328.08 -1.248 0.212365
## CompanyXiaomi 506.69 328.08 1.544 0.122740
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 643.8 on 1284 degrees of freedom
## Multiple R-squared: 0.1635, Adjusted R-squared: 0.1518
## F-statistic: 13.94 on 18 and 1284 DF, p-value: < 2.2e-16drop1(lmB, test = 'F')## Single term deletions
##
## Model:
## Price ~ Company
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 532160971 16873
## Company 18 104013991 636174961 17069 13.943 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(lmB)## Analysis of Variance Table
##
## Response: Price
## Df Sum Sq Mean Sq F value Pr(>F)
## Company 18 104013991 5778555 13.943 < 2.2e-16 ***
## Residuals 1284 532160971 414456
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1ls_Company = lsmeans(lmB,pairwise ~ Company,adjust = 'tukey')
#ls_Company$contrasts #too long to be printed
ls_Company$lsmeans## Company lsmean SE df lower.CL upper.CL
## Acer 627 63.4 1284 502.331 751
## Apple 1564 140.5 1284 1288.594 1840
## Asus 1104 51.2 1284 1003.692 1205
## Chuwi 314 371.7 1284 -414.885 1043
## Dell 1186 37.4 1284 1112.783 1259
## Fujitsu 729 371.7 1284 -0.182 1458
## Google 1678 371.7 1284 948.485 2407
## HP 1068 38.9 1284 991.475 1144
## Huawei 1424 455.2 1284 530.938 2317
## Lenovo 1086 37.4 1284 1013.099 1160
## LG 2099 371.7 1284 1369.818 2828
## Mediacom 295 243.3 1284 -182.362 772
## Microsoft 1612 262.8 1284 1096.699 2128
## MSI 1729 87.6 1284 1557.038 1901
## Razer 3346 243.3 1284 2868.781 3824
## Samsung 1413 214.6 1284 992.451 1834
## Toshiba 1268 92.9 1284 1085.517 1450
## Vero 217 321.9 1284 -414.065 849
## Xiaomi 1133 321.9 1284 501.972 1765
##
## Confidence level used: 0.95plot(ls_Company$lsmeans, alpha = .05)
lmC = lm(Price ~ TypeName, data=data)
summary(lmC)##
## Call:
## lm(formula = Price ~ TypeName, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1049.2 -381.7 -98.1 267.6 4367.6
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1282.40 50.01 25.642 < 2e-16 ***
## TypeNameGaming 448.98 63.07 7.119 1.79e-12 ***
## TypeNameNetbook -646.17 120.86 -5.347 1.06e-07 ***
## TypeNameNotebook -500.32 54.01 -9.263 < 2e-16 ***
## TypeNameUltrabook 265.83 63.60 4.180 3.12e-05 ***
## TypeNameWorkstation 997.96 113.74 8.774 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 550.1 on 1297 degrees of freedom
## Multiple R-squared: 0.383, Adjusted R-squared: 0.3806
## F-statistic: 161 on 5 and 1297 DF, p-value: < 2.2e-16drop1(lmC, test = 'F')## Single term deletions
##
## Model:
## Price ~ TypeName
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 392518380 16450
## TypeName 5 243656581 636174961 17069 161.02 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(lmC)## Analysis of Variance Table
##
## Response: Price
## Df Sum Sq Mean Sq F value Pr(>F)
## TypeName 5 243656581 48731316 161.02 < 2.2e-16 ***
## Residuals 1297 392518380 302636
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1ls_TypeName = lsmeans(lmC,pairwise ~ TypeName,adjust = 'tukey')
ls_TypeName$contrasts ## contrast estimate SE df t.ratio p.value
## 2 in 1 Convertible - Gaming -449 63.1 1297 -7.119 <.0001
## 2 in 1 Convertible - Netbook 646 120.9 1297 5.347 <.0001
## 2 in 1 Convertible - Notebook 500 54.0 1297 9.263 <.0001
## 2 in 1 Convertible - Ultrabook -266 63.6 1297 -4.180 0.0004
## 2 in 1 Convertible - Workstation -998 113.7 1297 -8.774 <.0001
## Gaming - Netbook 1095 116.5 1297 9.397 <.0001
## Gaming - Notebook 949 43.5 1297 21.821 <.0001
## Gaming - Ultrabook 183 55.0 1297 3.333 0.0114
## Gaming - Workstation -549 109.1 1297 -5.030 <.0001
## Netbook - Notebook -146 111.9 1297 -1.303 0.7833
## Netbook - Ultrabook -912 116.8 1297 -7.806 <.0001
## Netbook - Workstation -1644 150.1 1297 -10.951 <.0001
## Notebook - Ultrabook -766 44.3 1297 -17.304 <.0001
## Notebook - Workstation -1498 104.2 1297 -14.383 <.0001
## Ultrabook - Workstation -732 109.5 1297 -6.689 <.0001
##
## P value adjustment: tukey method for comparing a family of 6 estimatesls_TypeName$lsmeans## TypeName lsmean SE df lower.CL upper.CL
## 2 in 1 Convertible 1282 50.0 1297 1184 1381
## Gaming 1731 38.4 1297 1656 1807
## Netbook 636 110.0 1297 420 852
## Notebook 782 20.4 1297 742 822
## Ultrabook 1548 39.3 1297 1471 1625
## Workstation 2280 102.2 1297 2080 2481
##
## Confidence level used: 0.95plot(ls_TypeName$lsmeans, alpha = .05)
library(coefplot)
#library(forestmodel)
coefplot(lmC, intercept = FALSE)
par(mfrow = c(2,2))
plot(lmC)
#(not) normal distribution of residuals
par(mfrow=c(1,2))
boxplot(lmC$residuals)
qqnorm(lmC$residuals);qqline(lmC$residuals)
ad.test(lmC$residuals)##
## Anderson-Darling normality test
##
## data: lmC$residuals
## A = 22.667, p-value < 2.2e-16shapiro.test(lmC$residuals)##
## Shapiro-Wilk normality test
##
## data: lmC$residuals
## W = 0.89641, p-value < 2.2e-16#let's try again with the log correction
lmC_log = lm(log(Price) ~ TypeName, data=data)
summary(lmC_log)#R^2 increases##
## Call:
## lm(formula = log(Price) ~ TypeName, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.40971 -0.33589 0.00698 0.33215 1.96853
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.02648 0.04379 160.456 < 2e-16 ***
## TypeNameGaming 0.33865 0.05522 6.133 1.15e-09 ***
## TypeNameNetbook -0.91149 0.10583 -8.613 < 2e-16 ***
## TypeNameNotebook -0.49823 0.04729 -10.534 < 2e-16 ***
## TypeNameUltrabook 0.26648 0.05569 4.785 1.91e-06 ***
## TypeNameWorkstation 0.66479 0.09959 6.675 3.65e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4817 on 1297 degrees of freedom
## Multiple R-squared: 0.4061, Adjusted R-squared: 0.4038
## F-statistic: 177.4 on 5 and 1297 DF, p-value: < 2.2e-16drop1(lmC_log, test = 'F')## Single term deletions
##
## Model:
## log(Price) ~ TypeName
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 300.95 -1897.5
## TypeName 5 205.76 506.71 -1228.7 177.36 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(lmC_log)## Analysis of Variance Table
##
## Response: log(Price)
## Df Sum Sq Mean Sq F value Pr(>F)
## TypeName 5 205.76 41.152 177.36 < 2.2e-16 ***
## Residuals 1297 300.95 0.232
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1ls_TypeName_log = lsmeans(lmC_log,pairwise ~ TypeName,adjust = 'tukey')
ls_TypeName_log$contrasts ## contrast estimate SE df t.ratio p.value
## 2 in 1 Convertible - Gaming -0.3387 0.0552 1297 -6.133 <.0001
## 2 in 1 Convertible - Netbook 0.9115 0.1058 1297 8.613 <.0001
## 2 in 1 Convertible - Notebook 0.4982 0.0473 1297 10.534 <.0001
## 2 in 1 Convertible - Ultrabook -0.2665 0.0557 1297 -4.785 <.0001
## 2 in 1 Convertible - Workstation -0.6648 0.0996 1297 -6.675 <.0001
## Gaming - Netbook 1.2501 0.1020 1297 12.251 <.0001
## Gaming - Notebook 0.8369 0.0381 1297 21.970 <.0001
## Gaming - Ultrabook 0.0722 0.0481 1297 1.500 0.6644
## Gaming - Workstation -0.3261 0.0956 1297 -3.413 0.0087
## Netbook - Notebook -0.4133 0.0980 1297 -4.218 0.0004
## Netbook - Ultrabook -1.1780 0.1023 1297 -11.515 <.0001
## Netbook - Workstation -1.5763 0.1315 1297 -11.990 <.0001
## Notebook - Ultrabook -0.7647 0.0388 1297 -19.725 <.0001
## Notebook - Workstation -1.1630 0.0912 1297 -12.750 <.0001
## Ultrabook - Workstation -0.3983 0.0958 1297 -4.156 0.0005
##
## Results are given on the log (not the response) scale.
## P value adjustment: tukey method for comparing a family of 6 estimatesls_TypeName_log$lsmeans## TypeName lsmean SE df lower.CL upper.CL
## 2 in 1 Convertible 7.03 0.0438 1297 6.94 7.11
## Gaming 7.37 0.0336 1297 7.30 7.43
## Netbook 6.11 0.0963 1297 5.93 6.30
## Notebook 6.53 0.0179 1297 6.49 6.56
## Ultrabook 7.29 0.0344 1297 7.23 7.36
## Workstation 7.69 0.0894 1297 7.52 7.87
##
## Results are given on the log (not the response) scale.
## Confidence level used: 0.95plot(ls_TypeName_log$lsmeans, alpha = .05)
coefplot(lmC_log, intercept = FALSE)
par(mfrow = c(2,2))
plot(lmC_log)
#(not) normal distribution of residuals
par(mfrow=c(1,2))
boxplot(lmC_log$residuals)
qqnorm(lmC_log$residuals);qqline(lmC_log$residuals)
ad.test(lmC_log$residuals) #normal now!##
## Anderson-Darling normality test
##
## data: lmC_log$residuals
## A = 0.51757, p-value = 0.1886shapiro.test(lmC_log$residuals) #borderline now!##
## Shapiro-Wilk normality test
##
## data: lmC_log$residuals
## W = 0.99764, p-value = 0.05462A due vie
# Con interazione
lmC = lm(Price ~ Company*TypeName , data=data)
drop1(lmC, test="F")## Single term deletions
##
## Model:
## Price ~ Company * TypeName
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 320739568 16273
## Company:TypeName 25 29159364 349898932 16336 4.5602 1.181e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#summary(lmC) #FIXME: too long to be printed
lmC = lm(Price ~ Company+TypeName , data=data)
# type I effects A, B/A C/A,B
anova(lmC)## Analysis of Variance Table
##
## Response: Price
## Df Sum Sq Mean Sq F value Pr(>F)
## Company 18 104013991 5778555 21.123 < 2.2e-16 ***
## TypeName 5 182262038 36452408 133.246 < 2.2e-16 ***
## Residuals 1279 349898932 273572
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# type III effects A/B,C , B/A,C C/A,B
drop1(lmC, test="F")## Single term deletions
##
## Model:
## Price ~ Company + TypeName
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 349898932 16336
## Company 18 42619448 392518380 16450 8.6549 < 2.2e-16 ***
## TypeName 5 182262038 532160971 16873 133.2460 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(lmC)##
## Call:
## lm(formula = Price ~ Company + TypeName, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2147.6 -343.2 -81.9 243.1 4081.9
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 991.52 69.88 14.189 < 2e-16 ***
## CompanyApple 383.70 132.62 2.893 0.00388 **
## CompanyAsus 168.81 67.79 2.490 0.01290 *
## CompanyChuwi -180.68 306.47 -0.590 0.55559
## CompanyDell 350.73 60.52 5.796 8.56e-09 ***
## CompanyFujitsu 234.02 306.47 0.764 0.44525
## CompanyGoogle 497.17 309.44 1.607 0.10837
## CompanyHP 337.48 60.85 5.546 3.55e-08 ***
## CompanyHuawei 243.50 375.96 0.648 0.51731
## CompanyLenovo 322.12 60.24 5.348 1.05e-07 ***
## CompanyLG 918.50 309.44 2.968 0.00305 **
## CompanyMediacom -270.91 204.43 -1.325 0.18534
## CompanyMicrosoft 431.81 223.95 1.928 0.05406 .
## CompanyMSI 311.10 98.62 3.155 0.00165 **
## CompanyRazer 1996.14 207.24 9.632 < 2e-16 ***
## CompanySamsung 438.88 183.82 2.388 0.01710 *
## CompanyToshiba 601.45 92.12 6.529 9.52e-11 ***
## CompanyVero -277.55 266.70 -1.041 0.29821
## CompanyXiaomi 295.72 267.40 1.106 0.26896
## TypeNameGaming 426.29 65.51 6.507 1.10e-10 ***
## TypeNameNetbook -600.94 115.75 -5.192 2.42e-07 ***
## TypeNameNotebook -496.54 51.98 -9.552 < 2e-16 ***
## TypeNameUltrabook 188.98 63.81 2.962 0.00312 **
## TypeNameWorkstation 948.46 109.22 8.684 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 523 on 1279 degrees of freedom
## Multiple R-squared: 0.45, Adjusted R-squared: 0.4401
## F-statistic: 45.5 on 23 and 1279 DF, p-value: < 2.2e-16# contrasti
library(lsmeans)
ls=lsmeans(lmC, #FIXME: @Andrea, c'era lmB ma credo tu volessi scrivere lmC, in case check it
pairwise ~ TypeName ,
adjust="tukey")
ls$lsmeans## TypeName lsmean SE df lower.CL upper.CL
## 2 in 1 Convertible 1350 68.9 1279 1214 1485
## Gaming 1776 64.4 1279 1649 1902
## Netbook 749 115.9 1279 521 976
## Notebook 853 52.0 1279 751 955
## Ultrabook 1538 55.5 1279 1430 1647
## Workstation 2298 110.1 1279 2082 2514
##
## Results are averaged over the levels of: Company
## Confidence level used: 0.95# plot lsmeans and 95% confid int
plot(ls$lsmeans, alpha = .05)
# contrasts between predicted lsmeans
ls$contrasts## contrast estimate SE df t.ratio p.value
## 2 in 1 Convertible - Gaming -426 65.5 1279 -6.507 <.0001
## 2 in 1 Convertible - Netbook 601 115.7 1279 5.192 <.0001
## 2 in 1 Convertible - Notebook 497 52.0 1279 9.552 <.0001
## 2 in 1 Convertible - Ultrabook -189 63.8 1279 -2.962 0.0367
## 2 in 1 Convertible - Workstation -948 109.2 1279 -8.684 <.0001
## Gaming - Netbook 1027 114.5 1279 8.972 <.0001
## Gaming - Notebook 923 49.4 1279 18.671 <.0001
## Gaming - Ultrabook 237 61.1 1279 3.882 0.0015
## Gaming - Workstation -522 108.3 1279 -4.820 <.0001
## Netbook - Notebook -104 107.0 1279 -0.975 0.9258
## Netbook - Ultrabook -790 113.3 1279 -6.969 <.0001
## Netbook - Workstation -1549 143.8 1279 -10.774 <.0001
## Notebook - Ultrabook -686 46.5 1279 -14.754 <.0001
## Notebook - Workstation -1445 99.8 1279 -14.475 <.0001
## Ultrabook - Workstation -759 106.4 1279 -7.138 <.0001
##
## Results are averaged over the levels of: Company
## P value adjustment: tukey method for comparing a family of 6 estimates# if at least one contrast is significant, the variable
# is significant in the anova table # drop1 effects
# contrast among predicted lsmeans and overall lsmean
c= contrast(ls, method = "eff")
c## $lsmeans
## contrast estimate SE df t.ratio p.value
## 2 in 1 Convertible effect -77.7 47.9 1279 -1.623 0.1048
## Gaming effect 348.6 46.0 1279 7.583 <.0001
## Netbook effect -678.6 90.2 1279 -7.521 <.0001
## Notebook effect -574.2 31.8 1279 -18.032 <.0001
## Ultrabook effect 111.3 43.8 1279 2.542 0.0134
## Workstation effect 870.7 84.6 1279 10.287 <.0001
##
## Results are averaged over the levels of: Company
## P value adjustment: fdr method for 6 tests
##
## $contrasts
## contrast estimate SE df t.ratio
## 2 in 1 Convertible - Gaming effect -150.6 71.6 1279 -2.103
## 2 in 1 Convertible - Netbook effect 876.6 121.9 1279 7.192
## 2 in 1 Convertible - Notebook effect 772.2 51.2 1279 15.077
## 2 in 1 Convertible - Ultrabook effect 86.7 57.9 1279 1.498
## 2 in 1 Convertible - Workstation effect -672.8 74.0 1279 -9.093
## Gaming - Netbook effect 1302.9 123.6 1279 10.544
## Gaming - Notebook effect 1198.5 55.4 1279 21.649
## Gaming - Ultrabook effect 513.0 60.9 1279 8.416
## Gaming - Workstation effect -246.5 77.4 1279 -3.186
## Netbook - Notebook effect 171.2 107.0 1279 1.600
## Netbook - Ultrabook effect -514.3 110.5 1279 -4.655
## Netbook - Workstation effect -1273.7 119.6 1279 -10.649
## Notebook - Ultrabook effect -409.9 55.3 1279 -7.416
## Notebook - Workstation effect -1169.3 71.6 1279 -16.325
## Ultrabook - Workstation effect -483.8 84.4 1279 -5.730
## p.value
## 0.0411
## <.0001
## <.0001
## 0.1345
## <.0001
## <.0001
## <.0001
## <.0001
## 0.0018
## 0.1177
## <.0001
## <.0001
## <.0001
## <.0001
## <.0001
##
## Results are averaged over the levels of: Company
## P value adjustment: fdr method for 15 testslibrary(coefplot)
coefplot(lmC, intercept=FALSE) #FIXME: @Andrea, same goes here
ANOVA k way
lmK = lm(Price ~ Company+TypeName+SolidStateDisk , data=data)
summary(lmK)##
## Call:
## lm(formula = Price ~ Company + TypeName + SolidStateDisk, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2113.2 -301.6 -49.8 210.4 3862.4
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 689.90 67.44 10.230 < 2e-16 ***
## CompanyApple 514.90 122.54 4.202 2.83e-05 ***
## CompanyAsus 130.26 62.53 2.083 0.03744 *
## CompanyChuwi -16.66 282.67 -0.059 0.95300
## CompanyDell 280.98 55.97 5.020 5.88e-07 ***
## CompanyFujitsu 84.49 282.64 0.299 0.76505
## CompanyGoogle 404.40 285.26 1.418 0.15653
## CompanyHP 272.28 56.25 4.840 1.45e-06 ***
## CompanyHuawei 150.73 346.56 0.435 0.66368
## CompanyLenovo 235.35 55.81 4.217 2.65e-05 ***
## CompanyLG 825.73 285.26 2.895 0.00386 **
## CompanyMediacom -356.00 188.50 -1.889 0.05918 .
## CompanyMicrosoft 339.04 206.50 1.642 0.10087
## CompanyMSI 178.78 91.32 1.958 0.05047 .
## CompanyRazer 1868.90 191.19 9.775 < 2e-16 ***
## CompanySamsung 387.48 169.45 2.287 0.02238 *
## CompanyToshiba 436.72 85.60 5.102 3.87e-07 ***
## CompanyVero -113.54 246.04 -0.461 0.64456
## CompanyXiaomi 96.18 246.80 0.390 0.69680
## TypeNameGaming 398.61 60.40 6.599 6.05e-11 ***
## TypeNameNetbook -473.92 107.01 -4.429 1.03e-05 ***
## TypeNameNotebook -358.93 48.77 -7.360 3.28e-13 ***
## TypeNameUltrabook 113.04 59.02 1.915 0.05570 .
## TypeNameWorkstation 946.96 100.66 9.407 < 2e-16 ***
## SolidStateDiskTrue 470.33 31.17 15.089 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 482.1 on 1278 degrees of freedom
## Multiple R-squared: 0.5332, Adjusted R-squared: 0.5244
## F-statistic: 60.82 on 24 and 1278 DF, p-value: < 2.2e-16drop1(lmK, test="F") # type III SS## Single term deletions
##
## Model:
## Price ~ Company + TypeName + SolidStateDisk
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 296988657 16125
## Company 18 33990309 330978966 16230 8.1259 < 2.2e-16 ***
## TypeName 5 109128253 406116910 16523 93.9200 < 2.2e-16 ***
## SolidStateDisk 1 52910275 349898932 16336 227.6832 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1coefplot(lmK, intercept=FALSE)
Regressione lineare
lmA<-lm(Price ~ Frequenza , data=data)
summary(lmA)##
## Call:
## lm(formula = Price ~ Frequenza, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1467.6 -453.8 -119.6 327.6 4618.2
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -241.84 81.32 -2.974 0.003 **
## Frequenza 594.02 34.55 17.194 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 631.2 on 1301 degrees of freedom
## Multiple R-squared: 0.1852, Adjusted R-squared: 0.1845
## F-statistic: 295.6 on 1 and 1301 DF, p-value: < 2.2e-16plot(data$Frequenza,data$Price)
abline(lmA,col="red")
lmA<-lm(Price ~ Frequenza+Pixel+Ram , data=data)
summary(lmA)##
## Call:
## lm(formula = Price ~ Frequenza + Pixel + Ram, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1785.72 -257.23 -66.06 191.11 2791.53
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.076e+02 5.547e+01 -7.349 3.52e-13 ***
## Frequenza 2.549e+02 2.474e+01 10.306 < 2e-16 ***
## Pixel 1.329e-04 9.117e-06 14.575 < 2e-16 ***
## Ram 7.839e+01 2.658e+00 29.488 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 420.2 on 1299 degrees of freedom
## Multiple R-squared: 0.6395, Adjusted R-squared: 0.6386
## F-statistic: 768 on 3 and 1299 DF, p-value: < 2.2e-16coefplot(lmA, intercept=FALSE)
ANCOVA
lmK = lm(Price ~ Company+TypeName+SolidStateDisk+ Frequenza+Pixel+Ram , data=data)
summary(lmK)##
## Call:
## lm(formula = Price ~ Company + TypeName + SolidStateDisk + Frequenza +
## Pixel + Ram, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1838.5 -211.8 -28.2 169.3 1894.4
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.491e+02 6.691e+01 -2.229 0.02602 *
## CompanyApple 2.826e+02 9.043e+01 3.125 0.00182 **
## CompanyAsus 5.438e+01 4.587e+01 1.185 0.23609
## CompanyChuwi -7.683e+01 2.082e+02 -0.369 0.71213
## CompanyDell 1.124e+02 4.132e+01 2.720 0.00662 **
## CompanyFujitsu 5.168e+01 2.071e+02 0.250 0.80294
## CompanyGoogle 3.062e+02 2.105e+02 1.455 0.14602
## CompanyHP 2.045e+02 4.134e+01 4.947 8.54e-07 ***
## CompanyHuawei 5.510e+01 2.539e+02 0.217 0.82822
## CompanyLenovo 1.260e+02 4.108e+01 3.066 0.00221 **
## CompanyLG 6.759e+02 2.090e+02 3.235 0.00125 **
## CompanyMediacom -1.108e+02 1.392e+02 -0.796 0.42603
## CompanyMicrosoft 2.369e+02 1.515e+02 1.564 0.11807
## CompanyMSI 2.046e+02 6.686e+01 3.061 0.00225 **
## CompanyRazer 1.085e+03 1.428e+02 7.594 5.95e-14 ***
## CompanySamsung 9.436e+01 1.246e+02 0.757 0.44896
## CompanyToshiba 2.871e+02 6.306e+01 4.553 5.79e-06 ***
## CompanyVero 1.440e+01 1.811e+02 0.080 0.93663
## CompanyXiaomi -1.743e+01 1.808e+02 -0.096 0.92322
## TypeNameGaming -2.977e+01 4.812e+01 -0.619 0.53621
## TypeNameNetbook -1.142e+02 7.947e+01 -1.437 0.15105
## TypeNameNotebook -2.440e+02 3.642e+01 -6.700 3.11e-11 ***
## TypeNameUltrabook 9.405e+01 4.338e+01 2.168 0.03034 *
## TypeNameWorkstation 7.172e+02 7.500e+01 9.562 < 2e-16 ***
## SolidStateDiskTrue 1.997e+02 2.432e+01 8.212 5.28e-16 ***
## Frequenza 1.701e+02 2.320e+01 7.335 3.94e-13 ***
## Pixel 8.315e-05 8.292e-06 10.028 < 2e-16 ***
## Ram 6.541e+01 2.578e+00 25.368 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 352.9 on 1275 degrees of freedom
## Multiple R-squared: 0.7504, Adjusted R-squared: 0.7452
## F-statistic: 142 on 27 and 1275 DF, p-value: < 2.2e-16drop1(lmK, .~., test="F")## Single term deletions
##
## Model:
## Price ~ Company + TypeName + SolidStateDisk + Frequenza + Pixel +
## Ram
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 158760389 15315
## Company 18 13404444 172164833 15384 5.9806 4.092e-14 ***
## TypeName 5 35077529 193837917 15565 56.3413 < 2.2e-16 ***
## SolidStateDisk 1 8397143 167157532 15380 67.4372 5.281e-16 ***
## Frequenza 1 6698755 165459144 15367 53.7975 3.940e-13 ***
## Pixel 1 12521049 171281438 15412 100.5562 < 2.2e-16 ***
## Ram 1 80130237 238890626 15845 643.5236 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1ls=lsmeans(lmK,
pairwise ~ Company ,
adjust="tukey")
c= contrast(ls, method = "eff")
#c #FIXME: too long to be printeddata$LogPrice=NULL
data$Product=NULL
data$X=NULL
str(data)## 'data.frame': 1303 obs. of 15 variables:
## $ Company : Factor w/ 19 levels "Acer","Apple",..: 2 2 8 2 2 1 2 2 3 1 ...
## $ TypeName : Factor w/ 6 levels "2 in 1 Convertible",..: 5 5 4 5 5 4 5 5 5 5 ...
## $ Inches : num 13.3 13.3 15.6 15.4 13.3 15.6 15.4 13.3 14 14 ...
## $ ScreenResolution: Factor w/ 40 levels "1366x768","1440x900",..: 24 2 9 26 24 1 26 2 9 16 ...
## $ Cpu : Factor w/ 118 levels "AMD A10-Series 9600P 2.4GHz",..: 55 53 64 75 57 15 74 53 96 73 ...
## $ Ram : num 8 8 8 16 8 4 16 8 16 8 ...
## $ Memory : Factor w/ 39 levels "1.0TB HDD","1.0TB Hybrid",..: 5 3 17 30 17 27 16 16 30 17 ...
## $ Gpu : Factor w/ 110 levels "AMD FirePro W4190M",..: 59 52 54 10 60 18 61 52 98 62 ...
## $ OpSys : Factor w/ 9 levels "Android","Chrome OS",..: 5 5 6 5 5 7 4 5 7 7 ...
## $ Weight : num 1.37 1.34 1.86 1.83 1.37 2.1 2.04 1.34 1.3 1.6 ...
## $ Price : num 1340 899 575 2537 1804 ...
## $ Frequenza : num 2.3 1.8 2.5 2.7 3.1 3 2.2 1.8 1.8 1.6 ...
## $ Risoluzione : Factor w/ 15 levels "1366x768","1440x900",..: 11 2 4 13 11 1 13 2 4 4 ...
## $ Pixel : int 4096000 1296000 2073600 5184000 4096000 1049088 5184000 1296000 2073600 2073600 ...
## $ SolidStateDisk : Factor w/ 2 levels "False","True": 2 1 2 2 2 1 1 1 2 2 ...lm_full = lm(Price ~ ., data = data)
#summary(lm_full) #FIXME: wayyy too long to be printed, R^2 =0.9586
anova(lm_full, test="F")## Analysis of Variance Table
##
## Response: Price
## Df Sum Sq Mean Sq F value Pr(>F)
## Company 18 104013991 5778555 114.5882 < 2.2e-16 ***
## TypeName 5 182262038 36452408 722.8478 < 2.2e-16 ***
## Inches 1 6163570 6163570 122.2230 < 2.2e-16 ***
## ScreenResolution 36 108074619 3002073 59.5308 < 2.2e-16 ***
## Cpu 110 95329933 866636 17.1853 < 2.2e-16 ***
## Ram 1 34947028 34947028 692.9963 < 2.2e-16 ***
## Memory 35 17134540 489558 9.7079 < 2.2e-16 ***
## Gpu 88 34242874 389124 7.7163 < 2.2e-16 ***
## OpSys 6 3526085 587681 11.6537 1.198e-12 ***
## Weight 1 973 973 0.0193 0.8895
## Residuals 1001 50479311 50429
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1drop1(lm_full, test="F")## Single term deletions
##
## Model:
## Price ~ Company + TypeName + Inches + ScreenResolution + Cpu +
## Ram + Memory + Gpu + OpSys + Weight + Frequenza + Risoluzione +
## Pixel + SolidStateDisk
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 50479311 14370
## Company 14 6197922 56677232 14493 8.7789 < 2.2e-16 ***
## TypeName 5 3685931 54165241 14452 14.6183 7.50e-14 ***
## Inches 1 210134 50689445 14373 4.1669 0.04148 *
## ScreenResolution 23 5322877 55802188 14454 4.5892 8.09e-12 ***
## Cpu 88 16408116 66887427 14560 3.6974 < 2.2e-16 ***
## Ram 1 4481351 54960662 14479 88.8648 < 2.2e-16 ***
## Memory 34 10507055 60986365 14548 6.1281 < 2.2e-16 ***
## Gpu 88 30459868 80939179 14809 6.8638 < 2.2e-16 ***
## OpSys 6 3518495 53997806 14446 11.6286 1.28e-12 ***
## Weight 1 973 50480284 14368 0.0193 0.88953
## Frequenza 0 0 50479311 14370
## Risoluzione 0 0 50479311 14370
## Pixel 0 0 50479311 14370
## SolidStateDisk 0 0 50479311 14370
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#coefplot(lm_full, intercept=FALSE) #meglio di no ahah
par(mfrow=c(2,2))
plot(lm_full)## Warning: not plotting observations with leverage one:
## 13, 15, 18, 29, 34, 46, 84, 128, 160, 173, 178, 179, 205, 232, 267, 271, 299, 303, 324, 348, 388, 436, 438, 448, 457, 458, 466, 475, 501, 518, 520, 561, 564, 593, 611, 666, 671, 684, 689, 698, 702, 703, 718, 719, 724, 750, 756, 768, 777, 780, 781, 784, 808, 811, 817, 826, 827, 840, 848, 855, 902, 912, 922, 945, 946, 951, 954, 965, 977, 990, 1018, 1053, 1067, 1076, 1082, 1087, 1089, 1111, 1117, 1118, 1132, 1136, 1137, 1155, 1192, 1201, 1208, 1235
## Warning: not plotting observations with leverage one:
## 13, 15, 18, 29, 34, 46, 84, 128, 160, 173, 178, 179, 205, 232, 267, 271, 299, 303, 324, 348, 388, 436, 438, 448, 457, 458, 466, 475, 501, 518, 520, 561, 564, 593, 611, 666, 671, 684, 689, 698, 702, 703, 718, 719, 724, 750, 756, 768, 777, 780, 781, 784, 808, 811, 817, 826, 827, 840, 848, 855, 902, 912, 922, 945, 946, 951, 954, 965, 977, 990, 1018, 1053, 1067, 1076, 1082, 1087, 1089, 1111, 1117, 1118, 1132, 1136, 1137, 1155, 1192, 1201, 1208, 1235## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced
## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced
par(mfrow=c(1,1))
par(mfrow=c(1,2))
boxplot(lm_full$residuals)
qqnorm(lm_full$residuals);qqline(lm_full$residuals) # probably the correction would work pretty fine here
#tests
ad.test(lm_full$residuals)##
## Anderson-Darling normality test
##
## data: lm_full$residuals
## A = 19.821, p-value < 2.2e-16shapiro.test(lm_full$residuals)##
## Shapiro-Wilk normality test
##
## data: lm_full$residuals
## W = 0.94917, p-value < 2.2e-16library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:EnvStats':
##
## boxcoxboxcoxreg1<-boxcox(lm_full)
which.max(boxcoxreg1$y)## [1] 48lambda=boxcoxreg1$x[which.max(boxcoxreg1$y)]
lambda## [1] -0.1010101lm_full_t = lm(log(Price) ~ ., data = data)
par(mfrow=c(2,2))
plot(lm_full_t) #quite better## Warning: not plotting observations with leverage one:
## 13, 15, 18, 29, 34, 46, 84, 128, 160, 173, 178, 179, 205, 232, 267, 271, 299, 303, 324, 348, 388, 436, 438, 448, 457, 458, 466, 475, 501, 518, 520, 561, 564, 593, 611, 666, 671, 684, 689, 698, 702, 703, 718, 719, 724, 750, 756, 768, 777, 780, 781, 784, 808, 811, 817, 826, 827, 840, 848, 855, 902, 912, 922, 945, 946, 951, 954, 965, 977, 990, 1018, 1053, 1067, 1076, 1082, 1087, 1089, 1111, 1117, 1118, 1132, 1136, 1137, 1155, 1192, 1201, 1208, 1235
## Warning: not plotting observations with leverage one:
## 13, 15, 18, 29, 34, 46, 84, 128, 160, 173, 178, 179, 205, 232, 267, 271, 299, 303, 324, 348, 388, 436, 438, 448, 457, 458, 466, 475, 501, 518, 520, 561, 564, 593, 611, 666, 671, 684, 689, 698, 702, 703, 718, 719, 724, 750, 756, 768, 777, 780, 781, 784, 808, 811, 817, 826, 827, 840, 848, 855, 902, 912, 922, 945, 946, 951, 954, 965, 977, 990, 1018, 1053, 1067, 1076, 1082, 1087, 1089, 1111, 1117, 1118, 1132, 1136, 1137, 1155, 1192, 1201, 1208, 1235## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced
## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced
ad.test(lm_full_t$residuals) #not really##
## Anderson-Darling normality test
##
## data: lm_full_t$residuals
## A = 7.5169, p-value < 2.2e-16shapiro.test(lm_full_t$residuals) #not really##
## Shapiro-Wilk normality test
##
## data: lm_full_t$residuals
## W = 0.98478, p-value = 1.874e-10A look over outliers
library(car)## Loading required package: carData##
## Attaching package: 'car'## The following object is masked from 'package:EnvStats':
##
## qqPlot## The following object is masked from 'package:psych':
##
## logitinfluencePlot(lm_full,main="Influence Plot", sub="Circle size is proportial to Cook's Distance" )
## StudRes Hat CookD
## 13 NaN 1.00000000 NaN
## 15 NaN 1.00000000 NaN
## 434 2.124383 0.67519240 0.030955546
## 849 4.607995 0.04198279 0.003020118
## 1020 5.093056 0.04545388 0.003990584
## 1081 -4.381193 0.40114343 0.041814955#Cook's Distance
cooksd <- cooks.distance(lm_full_t)
cooksda=data.frame(cooksd)
summary(cooksd)## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 0.00000 0.00003 0.00013 0.00076 0.00053 0.04815 88# identify D values > 4/(n-k-1)
# Cook's D plot
cutoff <- 4/((nrow(data)-length(lm_full_t$coefficients)-2))
plot(lm_full_t, which=4, cook.levels=cutoff)
plot(cooksd, pch="*", cex=1, main="Influential Obs by Cooks distance") # plot cook's distance
abline(h = cutoff, col="red") # add cutoff line
text(x=1:length(cooksd)+1, y=cooksd, labels=ifelse(cooksd>4*mean(cooksd, na.rm=T),names(cooksd),""),
col="red")#add labels
#extract influencial obs
influential <- as.numeric(names(cooksd)[(cooksd > cutoff)]) # influential row numbers
influ=data.frame(data[cooksd > cutoff, ])
filtered_data <- data[ !(row.names(data) %in% c(influential)), ]
#Outlier rimossi
lm_full_t_no_OUTliers = lm(log(Price) ~ ., data = filtered_data)
par(mfrow=c(2,2))
plot(lm_full_t_no_OUTliers)## Warning: not plotting observations with leverage one:
## 15, 21, 32, 43, 148, 172, 196, 222, 288, 292, 312, 335, 408, 420, 430, 438, 439, 447, 456, 481, 485, 498, 500, 541, 544, 572, 589, 648, 651, 661, 666, 667, 679, 693, 694, 699, 724, 730, 742, 746, 751, 755, 758, 765, 791, 800, 801, 807, 814, 822, 829, 836, 863, 874, 883, 893, 914, 915, 919, 922, 932, 943, 944, 945, 957, 984, 1017, 1031, 1039, 1044, 1049, 1051, 1066, 1073, 1074, 1078, 1079, 1093, 1097, 1098, 1116, 1153, 1162, 1169, 1195, 1210
## Warning: not plotting observations with leverage one:
## 15, 21, 32, 43, 148, 172, 196, 222, 288, 292, 312, 335, 408, 420, 430, 438, 439, 447, 456, 481, 485, 498, 500, 541, 544, 572, 589, 648, 651, 661, 666, 667, 679, 693, 694, 699, 724, 730, 742, 746, 751, 755, 758, 765, 791, 800, 801, 807, 814, 822, 829, 836, 863, 874, 883, 893, 914, 915, 919, 922, 932, 943, 944, 945, 957, 984, 1017, 1031, 1039, 1044, 1049, 1051, 1066, 1073, 1074, 1078, 1079, 1093, 1097, 1098, 1116, 1153, 1162, 1169, 1195, 1210## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced
## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced
#summary(lm_full_t_no_OUTliers) #FIXME: too long to be printed, R^2=0.9727
ncvTest(lm_full_t_no_OUTliers)## Non-constant Variance Score Test
## Variance formula: ~ fitted.values
## Chisquare = 1.740444, Df = 1, p = 0.18708null = lm(log(Price) ~ 1, data = filtered_data)
full = lm(log(Price) ~ ., data = filtered_data)
lm_fit = stepAIC(null, scope = list(upper = full), direction = "both", trace = FALSE)
drop1(lm_fit, test = 'F')## Single term deletions
##
## Model:
## log(Price) ~ Cpu + Memory + OpSys + Gpu + TypeName + ScreenResolution +
## Company + Ram + Inches
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 32.161 -4053.8
## Cpu 84 15.5056 47.667 -3724.9 5.5789 < 2.2e-16 ***
## Memory 34 9.9754 42.136 -3780.6 8.8673 < 2.2e-16 ***
## OpSys 5 5.9181 38.079 -3850.5 35.7726 < 2.2e-16 ***
## Gpu 85 10.7290 42.890 -3860.2 3.8149 < 2.2e-16 ***
## TypeName 4 2.1668 34.328 -3979.5 16.3717 5.483e-13 ***
## ScreenResolution 28 4.7325 36.893 -3936.4 5.1082 5.348e-16 ***
## Company 14 3.0807 35.242 -3966.3 6.6506 4.801e-13 ***
## Ram 1 1.5499 33.711 -3996.4 46.8425 1.360e-11 ***
## Inches 1 1.0393 33.200 -4015.7 31.4100 2.720e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1